A pair-primary associative substrate with exact in-band episodic memory

We build a GPU-resident associative memory whose unit of representation is a relation: an ordered word pair with a signed binding strength, in place of the token. One rule drives all writes. On each exposure the binding state moves a fixed fraction of the way toward a ceiling that the world asserts. The rule admits an exact in-band episodic trace. Each binding carries six accumulators with distinct decay rates; together they form a sufficient statistic for the binding's exposure history. The ordered sequence of asserted targets follows by inverting the resulting generalized Vandermonde system. Count recovers exactly on all 4,640 live bindings (the complete live web, not a sample). Order recovers on the production bank to 1.5e-10 in float64 at depth six (2.4e-13 at depth three), and to 1.2e-3 through the production float32 kernels at four-step histories. The exactness claim is conditional: at least as many distinct, well-separated rates as history depth, and precision exceeding the conditioning (section 3.5). Those conditions collapse to one law, error ≈ ratio × cond(V) × precision (median ratio 0.03-0.05 over a six-order-of-magnitude sweep), so a unit's exactness envelope is computable before it is built (section 3.6). The rate set follows by minimizing cond(V), 867× better-conditioned than the reference (section 3.7). Pairing two rate bands lifts the exact depth from six to nine in float64, past the M ≥ k count wall (section 3.8). A shared-state control runs the identical law in a linear-attention layout. One binding alone on a shared state recovers exactly; at one thousand shared pairs the worst pair errs at 1.7e3 while the pair-primary arm holds at 1e-12 across all populations. The write-landing site, not the read geometry, carries the difference. Replaying the recorded input ledger rebuilds 9,272 of 9,296 nonzero live bindings, 99.7 percent. The two theorems supply state-only inversion for the organ-backed quantities, count and order.